Before discussing further some particular choices for the lapse function (REF ) we want to extend the techniques known for kinetic energy Lagrangians, see e.g. [1]}, [2]} and references therein, to our case including a potential energy. In this scenario one makes the following choice \(d{\mathcal {T}}=N(y)dy\) , where the function
\(N(y)\)  has to be determined and \(y=x^2\)  is the second coordinate. Now (cf. (REF ), \(\dot{()}\equiv \frac{d}{d y}\)  and \(\dot{y}=1\) ):
\(\ddot{x}\: +\: \widehat{G}^1_{11}\dot{x}^2 \:+\: \widehat{G}^1_{12}\dot{x}+ \widehat{G}^1_{22}= -N^2 m^{1j}\partial _j V_\text{MSS}, \\\widehat{G}^2_{11}\dot{x}^2\: +\widehat{G}^2_{12}\dot{x}+\widehat{G}^2_{22} = -N^2 m^{2j}\partial _j V_\text{MSS},\) 
